y'(x)=sqrt2y(x)+7, y(4)=1

Solve the seperable diff eq

To solve the separable differential equation y'(x) = √2y(x) + 7, we can follow these steps:

Step 1: Separate the variables
To separate the variables, we need to move all the terms involving y to one side of the equation:
y'(x) - √2y(x) = 7

Step 2: Integrate both sides
Integrate both sides of the equation with respect to x:
∫ (y'(x) - √2y(x)) dx = ∫ 7 dx

The left side can be split into two integrals:
∫ y'(x) dx - √2 ∫ y(x) dx = ∫ 7 dx

Integrating y'(x) with respect to x gives us y(x), and integrating 7 with respect to x gives us 7x:
y(x) - √2 ∫ y(x) dx = 7x + C

Step 3: Solve for y(x)
To solve for y(x), we need to evaluate the integral on the left side of the equation.

Let's assume that the integral of y(x) is F(y(x)) + C_1, where F(y(x)) is an antiderivative of y(x) and C_1 is the constant of integration.

Substituting this into the equation, we get:
y(x) - √2 (F(y(x)) + C_1) = 7x + C

Step 4: Solve the integral
To find the integral ∫ y(x) dx, we can use any integration technique. The specific integration technique will depend on the form of the function y(x). Without more information about y(x), it's not possible to provide an exact result at this point.

Step 5: Substitute initial condition
Use the initial condition y(4) = 1 to find the value of the constant C. Substitute x = 4 and y = 1 into the equation:
1 - √2 (F(1) + C_1) = 7(4) + C

Solve this equation to find the value of C.

By following these steps and finding the appropriate antiderivative F(y(x)), you can solve the separable differential equation y'(x) = √2y(x) + 7.